A deletion game on hypergraphs

Two players alternately select either a vertex or an edge of a hypergraph H, deleting it together with all the edges containing the selected vertex or edge. The player first unable to move loses and the other player wins. We analyze the game for simple classes of hypergraphs, and discuss various related questions.     

Bounds on the localization number

We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph G is called the localization number and is written as ζ(G). We settle a conjecture of Bosek et al by providing an upper bound on the chromatic number as a function of the localization number. In particular, we show that every graph with ζ(G) ≤ k has degeneracy less than 3^k and, consequently, satisfies χ(G) ≤ 3^ζ(G). We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.     

A note on the edge cover number and independence number in hypergraphs

Consider a hypergraph of rank r>2 with m edges, independence number α and edge cover number ρ. We prove the inequality

     

Sparse hypergraphs and pebble game algorithms

A hypergraph G=(V,E) is (k,ℓ)-sparse if no subset V^′V spans more than k|V^′|−ℓ hyperedges. We characterize (k,ℓ)-sparse hypergraphs in terms of graph theoretic, matroidal and algorithmic properties. We extend several well-known theorems of Haas, Lovász, Nash-Williams, Tutte, and White and Whiteley, linking arboricity of graphs to certain counts on the number of edges. We also address the problem of finding lower-dimensional representations of sparse hypergraphs, and identify a critical behavior in terms of the sparsity parameters k and ℓ. Our constructions extend the pebble games of Lee and Streinu [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs.     

Bounds for the Graham–Pollak theorem for hypergraphs

Let $f_r\left(n\right)$ represent the minimum number of complete $r$-partite $r$-graphs required to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. The Graham–Pollak theorem states that $f_2\left(n\right)=n?1$. An upper bound of $(1+o\left(1\right))\left(\genfrac{}{}{0ex}{}{n}{?\frac{r}{2}?}\right)$ was known. Recently this was improved to $\frac{14}{15}(1+o\left(1\right))\left(\genfrac{}{}{0ex}{}{n}{?\frac{r}{2}?}\right)$ for even $r\ge 4$. A bound of $\left[\frac{r}{2}{\left(\frac{14}{15}\right)}^{\frac{r}{4}}+o\left(1\right)\right](1+o\left(1\right))\left(\genfrac{}{}{0ex}{}{n}{?\frac{r}{2}?}\right)$ was also proved recently. Let $c_r$ be the limit of $\frac{f_r\left(n\right)}{\left(\genfrac{}{}{0ex}{}{n}{?\frac{r}{2}?}\right)}$ as $n\to \infty$. The smallest odd $r$ for which $c_r<1$ that was known was for $r=295$. In this note we improve this to $c_{113}<1$ and also give better upper bounds for $f_r\left(n\right)$, for small values of even $r$.     

On the chromatic number of Kneser hypergraphs

We give a simple and elementary proof of Kríz's lower bound on the chromatic number of the Kneser -hypergraph of a set system .

     

Critical hypergraphs for the weak chromatic number

Instead of removing a vertex or an edge from a hypergraph H, one may add to some edges of H new vertices (not necessarily belonging to V_H). The weak chromatic number of H tends to drop by this operation. This suggests the definition of an order relation ≥ on the set $\text{S}$ of all Sperner hypergraphs on a universal set V of vertices. The corresponding criticality study leads to unifying and interesting results: reconstruction of critical hypergraphs and two general characterizations of k-chromatic critical hypergraphs (k ≥ 3), from which a special characterization of 3-chromatic critical hypergraphs can be derived.     

The circular chromatic number of hypergraphs

A generalization of the circular chromatic number to hypergraphs is discussed. In particular, it is indicated how the basic theory, and five equivalent formulations of the circular chromatic number of graphs, can be carried over to hypergraphs with essentially the same proofs.     

A lower bound on the independence number of arbitrary hypergraphs

We present a lower bound on the independence number of arbitrary hypergraphs in terms of the degree vectors. The degree vector of a vertex v is given by d(v) = (d₁(v), d₂(v), …) where d_m(v) is the number of edges of size m containing v. We define a function f with the property that any hypergraph H = (V, E) satisfies α(H) ≥ Σ_v∈V f(d(v)). This lower bound is sharp when H is a match, and it generalizes known bounds of Caro/Wei and Caro/Tuza for ordinary graphs and uniform hypergraphs. Furthermore, an algorithm for computing independent sets of size as guaranteed by the lower bound is given. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 213–221, 1999